For an odd integer r>0 and n>r, we introduce a notion of weakly r-separated collections subsets [n]={1,2,…,n}. When r=1, this corresponds to the concept weak separation introduced by Leclerc Zelevinsky. In paper, extending results due Leclerc-Zelevinsky, develop geometric approach establish number nice combinatorial properties maximal (such as exact upper bound on size collections, mutations ru...

We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster’s operad for weak ω-categories.

We decribe the correspondence between normalised ω-operads in the sense of [1] and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category. Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised ωoperad, and categories enri...